Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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How to prove $\frac{(2n)!(2m)!}{n!m!(n+m)!}$ is an integer by strictly using my method?

I have to prove that $$\frac{(2n)!(2m)!}{n!m!(n+m)!}$$ is always an integer. I already have seen the same question here-Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}...
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2answers
83 views

Prove that $\log(n!)\leq(\log(n))!$

Prove that $\log(n!)\leq(\log(n))!$ My attempt: I read somewhere that $n\leq\log(n!)\leq(\log(n))!$. But when I used calculator $\log(n!)$ can not be less than or equal to $(\log(n))!$. ...
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0answers
20 views

Function order of Logarithms

How can I use Stirling's approximation to trying to find the function order of $ceil(log(logx))!$ ? My main goal is to finding it's order of complexity but my main issue is that I'm not sure on how to ...
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2answers
64 views

Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
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1answer
91 views

Find last 5 significant digits of 2017!

Since there are less powers of $5$ than of $2$ and since $10 = 2 \cdot 5$, I counted the number of zeros in $2017!$: $\left \lfloor{ \frac{2017}{5^1}}\right \rfloor + \left \lfloor{ \frac{2017}{5^2}}\...
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1answer
61 views

Conjecture concerning modular arithmetic

Below $0\notin\mathbb N$. I want a proof or a counter-example of the following (corrected) conjecture: Suppose $p$ is the smallest prime dividing $n\in\mathbb N$ and suppose $kn+ap=m!$, where $...
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1answer
311 views
+100

Conjecture about primes and the factorial

Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...
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1answer
26 views

Perfect square from a multiple of factorials

I have a problem with this question John writes the number 1!, 2!, 3!, ... , 199!, 200! on a whiteboard. John then erases one of the numbers. John then multiplied the remaining 199 numbers. He found ...
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4answers
171 views

Why does $(128)!$ equal the product of these binomial coefficients $128! = \binom{128}{64}\binom{64}{32}^2 \dots \binom21^{64}$?

I'm working through some combinatorics practice sets and found the following problem that I can't make heads or tails of. It asks to prove the following: $$128! = \binom{128}{64}\binom{64}{32}^2\...
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2answers
63 views

Restrictions on Factorial Usage

I had always understood that the factorial n! was defined as $$\prod_{k=1}^nk$$ However, this leaves several questions: Why does 0! exist?* By extension, why can't you take the factorial of a ...
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0answers
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Can Euler's generalization of factorial be done for double factorial?

Can Euler's generalization of factorial be done for double factorial? Euler's generalization of factorial to non-integer values is $t! =\lim_{n \to \infty} \dfrac{n!n^t}{\prod_{k=1}^n(t+k)} $. I ...
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2answers
55 views

How to find the Summation of series of Factorials?

$$1\cdot1!+2\cdot2!+\cdots+x\cdot x! = (x+1)!−1$$ I don't understand what's happening here. The given sum of factorials is generalized into a single term. Could somebody please help me finding the ...
12
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1answer
113 views

The divergent sum of alternating factorials

So I came across this exposition of a paper by Euler here where Euler is trying to sum the divergent sum: $$s = 1 - 1 + 2! - 3! + 4! \dots = \sum_{k\geq 0}(-1)^k k!.$$ There are a couple of questions ...
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0answers
23 views

Tight lower bound on falling factorial

I have the term $$p=\left(\frac{1}{n-b}\right)^a\cdot[n]_a, 0<b<a,\ b \in \mathbb{R} \text{ fixed}$$ and want to find a tight lower bound such that I will then be able to solve for n. For ...
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2answers
50 views

Expanding a factorial

Can you explain me how we got this identity? $$\frac{1}{(3n)!}$$ the same as $$\frac{(3n)!}{(3n+3)!}$$ I have been trying to expand, but didn't get the same. Thanks.
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1answer
75 views

Maybe this inequality holds? $x!-y!>x^n$?

Let $x,y,n$ be postive integers such that $x\ge 2y,y>n,n>3$ I conjectured that $$\color{red}{x!-y!\ge x^n}$$ Now, I claim that $$\color{red}{x!-y!=y![x(x-1)(x-2)\cdots(y+1)-1]\ge (n+1)![x(x-1)(...
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4answers
1k views

Evaluate $\frac{0!}{4!}+\frac{1!}{5!}+\frac{2!}{6!}+\frac{3!}{7!}+\frac{4!}{8!}+\cdots$

$$\frac{0!}{4!}+\frac{1!}{5!}+\frac{2!}{6!}+\frac{3!}{7!}+\frac{4!}{8!}+\frac{5!}{9!}+\frac{6!}{10!}+\cdots$$ This goes up to infinity. Trying finite cases may help. My Attempt:It seems that it is ...
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1answer
43 views

Is my proof to proove that $\frac{n!}{p_1!p_2!p_3!…p_m!}\in \mathbb{N}$ valid?

I wish to prove that $\frac{n!}{p_1!p_2!p_3!....p_m!}$ is an integer, were, $p_1+p_2+p_3+...+p_m=n$ and $p_i, n\in \mathbb{N}$. Pleace do check the validity of my proof. Let us consider the following ...
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3answers
80 views

Limit of $\binom{n}{k}/n^k$

How to prove that, $n$ approach infinity and $0\leq k\leq n$ : $$ \lim_{n\to\infty} \dfrac{1}{k!}\times\dfrac{n!}{(n-k)!\times n^k} = \dfrac{1}{k!}$$ I have no idea to begin the proof... Thanks in ...
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1answer
58 views

Is $S_i(1,2,\dots,p-2) \equiv 1 \pmod{p}$ for all values of $i$ whenever $p$ is prime?

Let $S_i(x_1,x_2,\dots,x_n)$ denote the $i$th elementary symmetric polynomial in $n$ variables. Is $S_i(1,2,\dots,p-2) \equiv 1 \pmod{p}$ for all values of $i$ from $0$ to $(p-2)$ whenever $p$ is ...
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3answers
65 views

Infinite Sum of Falling Factorial and Power

According to Mathematica, $$\sum_{k=0}^\infty \frac{(G+k)_{G-1}}{2^k}=2(G-1)!(2^{G}-1)$$ where $$(G+k)_{G-1}=\frac{(G+k)!}{(G+k-G+1)!}=\frac{(G+k)!}{(k+1)!}$$ is the falling factorial. I would ...
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4answers
137 views

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{n}{\left(n+1\right)!} = 1-\frac{1}{\left(n+1\right)!}$ So I proved the base case where $n=1$ and got $\frac{1}{2}...
3
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1answer
187 views

Show that $101!+1$ is not prime number [closed]

Show that $101!+1$ is not prime number. How many ways exist to do it?
21
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2answers
1k views

Is $\sqrt{n!}$ a natural number?

I'm new here (on Mathematics Stack Exchange). Also, I'm a 10th grade student not a math expert. So, my question is whether, $$\sqrt {n!} $$ comes in the set of the Natural Numbers. There were ...
3
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2answers
66 views

How to find the summation of this infinite series: $\sum_{k=1}^{\infty} \frac{1}{(k+1)(k-1)!}(1 - \frac{2}{k})$

I've been trying to figure out the following sum for a while now: $$\sum_{k=1}^{\infty} \frac{1}{(k+1)(k-1)!}\left(1 - \frac{2}{k}\right)$$ I'm pretty sure that this doesn't evaluate to $0$. ...
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3answers
608 views

Find $\lim_{n \to +\infty} \frac{1}{n}\sqrt[n]{\frac{(2n)!}{n!}} $ using Riemann integral

Wonder how to determine this limit by the use of Riemann integral. The limit is as follows: $$\lim_{n \to +\infty} \frac{1}{n}\sqrt[n]{\frac{(2n)!}{n!}} $$ My instructor told me that the usage of ...
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3answers
56 views

Prove this for any $k>0$

Prove that $k!>(\frac{k}{e})^{k}$. It is known that $e^{k}>(1+k)$. So if we multiply $k!$ on both sides, we get $k!e^{k}>(k+1)!$. Also $k^k>k!$. Now how to proceed ?
9
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1answer
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Calcule $\gcd(0!+1!+\ldots+n!, (n+1)!)$

I have to compute $d=\gcd(0!+1!+\ldots+n!, (n+1)!)$, so let $a=0!+1!+\ldots+n!$ and $b=(n+1)!$. Then: $a=0!+1!+\ldots+n!=3!+0!+1!+2!+4!+...+n!=6+4+4!+...+n! \equiv 2 \mod 4$ Thus, $a$ and $b$ are ...
2
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1answer
38 views

Irrationality of the decimal fraction

Surfing the internet I bumped into a very interesting problem, which I tried to solve, but got no results. The problem is following: let $h_n$ be the most right non-zero digit of $n!$, for example, $...
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3answers
408 views

Combinatorial formulas and interpretations

I found that $$ \sum_{j=0}^{s}(n-s+j)!\binom{s}{j}(s-j)! =s! \sum_{j=0}^{s} \frac{(n-s+j)!}{j!} = \frac{(n+1)!}{n+1-s}$$ I proved this formula with induction, but I was wondering if there is a (...
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1answer
73 views

Find the number of zeros at the end of $n!!$. [closed]

Can anyone give me a generalized way to find the number of zeroes at the end of $n!!$ ?
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3answers
171 views

Find the missing digits in the expansion of $34!$ [closed]

If $34!=295232799cd96041408476186096435ab000000$ then find the value of $a,b,c$ and $d.$ My Attempt: I can find that $b=0$ because it has seven five integers. Note: calculator is not allowed.
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3answers
138 views

Expanding $\frac{\Gamma(n)}{\Gamma(n-k)}$ as a polynomial

I want to expand $\frac{\Gamma(n)}{\Gamma(n-k)}$ as a polynomial, where $\Gamma$ is the gamma function. For $k\in\mathbb{N}$, it can be "simplified" as $$\frac{\Gamma(n)}{\Gamma(n-k)}=(n-1)(n-2)(n-3)...
3
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1answer
71 views

Exponent analog to the factorial function

Triangular numbers can be discovered by taking any number $n$, and adding $$\sum_{i=0}^n i = n + (n - 1) + (n - 2) ... 1 = \frac{n(n + 1)}{2}$$ These numbers can be generalized by putting any real ...
18
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5answers
901 views

A strange combinatorial identity [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
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0answers
43 views

Proving primality of $p$ without making any calculation involving $p$ directly

Wilson's Theorem states that a positive integer $p > 1$ is prime if and only if $(p-1)! \equiv -1 \pmod p$, showing a relationship between factorials and prime numbers. Finding it curious, today I ...
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0answers
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multifactorial of non-integer

I want to calculate 12.1!!!!!! , Just for curiosity. (One of my friend texted the term to me for some complex reason.) I searched for multifactorial in terms of gamma function or equation, and found ...
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0answers
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A new formula relating the factorial and Riemann Zeta function resp. Bernoulli numbers?

I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers: $$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 \...
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2answers
36 views

Mathematical Induction with series and factorials.

I wish to show the following $$ a_{n}=\sum_{k=0}^{n}\frac{1}{(2k+1)!(2(n-k)+1)!}=\sum_{k=0}^{n+1}\frac{1}{(2k)!(2(n+1-k))!}=b_{n+1} $$ for $n\geq0$ and wish to do it using induction. I've shown it ...
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2answers
53 views

Factoring the factorials

Just for the fun of it, I've started factoring $n!$ into its prime divisors, and this is what I got for $2\leq n\leq20$: $$\begin{align} 2! &= 2^\color{red}{1} &S_e=1\\ 3! &= 2^\color{red}...
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3answers
735 views

How to prove the sum of combination is equal to $2^n - 1$

One of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...
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0answers
108 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
2
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2answers
63 views

Factorial Divisibility

Let $a$ and $b$ be positive integers greater than one. With that in mind, $$(a \cdot b)!$$ is not necessarily divisible by: a) $$a!^b$$ b) $$b!^a$$ c) $$a! \cdot b!$$ d) $${2}^{ab}$$ By brute-...
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3answers
54 views

Proving that $\frac{(N+p-1)!}{p!(p-1)!N!}$ is an integer?

Consider the quantity: $$\frac{(N+p-1)!}{p!(p-1)!N!}$$ where $N$ and $p$ are positive integers. How can we show that this is always an integer (which I believe has to be the case since it represents ...
2
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1answer
77 views

Is there an equivalent to the Bertrand's postulate between factorials and primorials?

As the title explains, I am trying to know if there is a definition about the upper limit to find the first primorial $p_i\#$ (following the definition at OEIS) existing after a given factorial $n!$ (...
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4answers
29 views

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$ I think I'm having a bit of algebra problem with this proof. Here is my work thus far:$$\frac{n!}{k!(n-k)!}+...
1
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2answers
48 views

Suppose a coin in tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences…

Suppose a coin is tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences are there in which there are at least $5$ tails in a row? I know this is Permutation with repetition. My ...
0
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1answer
36 views

Trying to Express A Factorial As A Polynomial

I'd like to express the following as a polynomial. $$(a-1)(a-2)(a-3) . . . (a-b)$$ where $b<a$ I'm currently working on it now, but wanted to see if anyone's already done it, or already know what ...
1
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3answers
49 views

Demonstrating that 1! is = 1

The problem with this explanation is that it's using n = 2 instead of n = 1. Please read the explanation I found on "Math Forum - Ask Dr. Math" ( http://mathforum.org/library/drmath/view/57128.html ). ...
5
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5answers
106 views

For integer $n>2$, $(n!)^2 > n^n$ [duplicate]

Problem: For integer $n>2$, show that $(n!)^2 > n^n$ My attempt: I tried using induction. For $n=3$, the given condition is satisfied. Let us suppose $k!^2>k^k$ for some $k\geq3$. Then, $(...